Answer
Limit does not exist
Work Step by Step
Use polar-coordinates:
$x= r \cos \theta , y = r \sin \theta \\r^2=x^2+y^2$
Now, $$ \lim\limits_{(x,y) \to (0,0) } f(x,y)=\lim\limits_{(x,y) \to (0,0) } \dfrac{y^2}{x^2+y^2}\\=\lim\limits_{r \to 0} \cos (\dfrac{ r^2 \sin ^2 \theta}{r^2})\\ =\sin^2 \theta $$
We see that $ \lim\limits_{(x,y) \to (0,0) } f(x,y)=\sin^2 \theta $ is not a unique value.
Thus, we conclude that $ \lim\limits_{(x,y) \to (0,0) } f(x,y)=\lim\limits_{(x,y) \to (0,0) } \dfrac{y^2}{x^2+y^2}=DNE$
